Jesus would want us to follow his teachings in that verse, and elsewhere as well. If you don't mean to say you see his words in that verse as hate, why did you bring up hate about that?
There are two things one can mean by hate, the first is malice and the desire to destroy someone, the second is choosing someone, or something over them (hate by comparison). The first is the kind we shouldn't have for people as Christians.
type 1
"You must not harbor hatred against your brother in your heart. Directly rebuke your neighbor, so that you will not incur guilt on account of him. Do not seek revenge or bear a grudge against any of your people, but love your neighbor as yourself." (Leviticus 19:17-18)
type 2
“If anyone comes to me and does not hate father and mother, wife and children, brothers and sisters—yes, even their own life—such a person cannot be my disciple." (Luke 14:26)
A Christian should love God above everyone else (see Matthew 22:37-40). Many of the LGBT want Christians to choose validating their choices over the Christian's own love for God. The "hate," as they call it, is choosing God over them, not malice, or a desire to destroy them.
By love the Bible doesn't mean universal affirmation of behavior to make someone feel good, or accepted. It means seeking the good of the other, which can, at times, involve telling them no, or some truth they find harsh. It could even involve rebuking them for their behavior, as the first verse above says, instead of remaining silent. Hate that Christians are to avoid has nothing to do with non-approval of behavior. They shouldn't have malice-hate for people, but they are supposed to hate sin, and evil.
"Love must be sincere. Hate what is evil; cling to what is good." (Romans 12:9 )
Historically allowing same-sex relationships in some places isn't the same as pretending that they are marriages. You say nobody's redefining things, but saying marriage isn't by definition opposite-sex is doing exactly that. Judges legislating from the bench instead of letting people vote on it is forcible redefinition to a tee.
If God made marriage, and made it rooted in the union of male and female, as Jesus said, then someone saying it can be same-sex is not speaking the truth. Marriage is something real, and defined, not just a perception, or feeling people have about themselves, or even an invention by human government. God made it, and defined it.